Recent improvements in parallel imaging performance have been driven by the use of ever-greater numbers of independent surface coils placed so as to maximize the ability to unwrap the aliasing that occurs along a linear phase encode direction, and this work is proceeding with ever diminishing returns due to both the greatly increasing expense associated with all of the receiver channels, and due to coil coupling problems which dominate as the size of the coil elements decreases. This work introduces a new approach to more efficient parallel imaging (with fewer coils) using novel gradient encoding schemes that provide optimally complementary spatial information relative to that provided by the receiver coils. This approach introduces nonlinear gradient encoding using 1st and 2nd order spherical harmonics. A formalism is introduced for calculating the optimal set of encoding gradient for a given receiver coil array. For each acquired echo, a different gradient from this complementary set is applied during the readout and conventional phase encoding is discarded. The advantage of this approach lies in the complementary spatial encoding contributed by both the receiver coils and the gradient encoding scheme. A further advantage arises with the use of frequency encoding gradients that span the xy-plane and thus provides 2D information in a single echo. With this type of read gradient oversampling in the readout significantly improves the accelerated acquisition with no time penalty. With conventional Cartesian sampling there is no benefit, with respect to aliasing, to over-sampling the read-out. This project is aimed at building a high-speed shield gradient insert capable of generating a series of 2nd order spherical harmonic gradient shapes. This gradient set will be used for proof of principle, and we will simultaneously further develop the theory of Null Space Imaging (how to calculate the optimal gradient set) as well as the reconstruction methodology. The end product from this work would be both a validation of a new methodology for performing highly accelerated parallel imaging, allowing acceleration factors of 8 or more with as few as 8 receiver coils and a design for a larger gradient insert coil capable of imaging the human head. The project carries a natural benefit to public health through acceleration of the data acquisition process, thus allowing for higher resolution, or more thorough examinations on existing platforms, through the application of this Null Space Imaging approach. The proposal also fits with the goals of NIBIB for developments in accelerated parallel MR imaging.